The Intermediate Value Theorem

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Section 5.5 The Intermediate Value
Theorem Rolles Theorem The Mean Value Theorem
3.6
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Intermediate Value Theorem (IVT)
If f is continuous on a, b and N is a value
between f(a) and f(b), then there is at least one
point c between a and b where f takes on the
value N.
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Rolles Theorem
If f is continuous on a, b, if f(a) 0, f(b)
0, then there is at least one number c on (a, b)
where f (c ) 0
slope 0
f (c ) 0
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Given the curve
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If f is continuous on a, b and differentiable
on (a, b)
Conclusion Slope of Secant Line Equals Slope of
Tangent Line
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f(0) -1 f(1) 2
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Verify..f(0) 0 0 0 f(1) 1 1
0
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Find the value(s) of c that satisfy the Mean
Value Theorem for
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Note The Mean Value Theorem requires the
function to be continuous on -4, 4 and
differentiable on (-4, 4). Therefore, since f(x)
is discontinuous at x 0 which is on -4, 4,
there may be no value of c which satisfies the
Mean Value Theorem
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(No Transcript)
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f(x) is continuous and differentiable on -2, 2
On the interval -2, 2, c 0 satisfies the
conclusion of MVT
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f(x) is continuous and differentiable on -2, 1
On the interval -2, 1, c 0 satisfies the
conclusion of MVT
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Since f(x) is discontinuous at x 2, which is
part of the interval 0, 4, the Mean Value
Theorem does not apply
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f(x) is continuous and differentiable on -1, 2
c 1 satisfies the conclusion of MVT
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CALCULATOR REQUIRED
f(3) 39 f(-2) 64
For how many value(s) of c is f (c ) -5?
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